On isolated sets of solutions of some two-point boundary value problems

✍ Scribed by Roman Srzednicki

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 151 KB
• Volume: 54
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(03)00103-2

✦ Synopsis

We present a geometric approach to the question of existence of solutions of the two-point boundary value problem ẋ = f(t; x);

x(a) ∈ P;

x(b) ∈ Q;

where P and Q are submanifolds of the phase space. For an isolated set K of initial values of solutions of the problem, we associate the intersection index -(f; K), an element of Z (or of Z2 if some of the submanifolds is not orientable) satisfying the solvability (i.e. -(f; K) = 0 implies K = ∅), additivity and continuation invariance properties. We prove a theorem on calculation of -(f; K) if K is naturally generated by an isolated segment which is concordant, in some way, with the considered problem. As an application, we provide another proof of the classical Bernstein-Nagumo Theorem on existence of solutions of some second-order boundary value problems. Other applications refer to problems associated with ÿrst-order planar equations.